def set_constants(self):
if self.xcutoff is None:
trunc_error = 0.1 * self.abseps
self.nc1c2 = max(1, self.nc1c2)
xcut = abs(invnorm(trunc_error / (self.nc1c2 * 2)))
self.xcutoff = max(min(xcut, 8.5), 1.2)
# self.abseps = max(self.abseps- truncError,0);
# self.releps = max(self.releps- truncError,0);
if self.method > 0:
names = [
"method",
"xcscale",
"abseps",
"releps",
"coveps",
"maxpts",
"minpts",
"nit",
"xcutoff",
"nc1c2",
"quadno",
"xsplit",
]
constants = [getattr(self, name) for name in names]
constants[0] = mod(constants[0], 10)
rindmod.set_constants(*constants) # @UndefinedVariable
def set_constants(self):
if self.xcutoff is None:
trunc_error = 0.1 * self.abseps
self.nc1c2 = max(1, self.nc1c2)
xcut = abs(invnorm(trunc_error / (self.nc1c2 * 2)))
self.xcutoff = max(min(xcut, 8.5), 1.2)
# self.abseps = max(self.abseps- truncError,0);
# self.releps = max(self.releps- truncError,0);
if self.method > 0:
names = ['method', 'xcscale', 'abseps', 'releps', 'coveps',
'maxpts', 'minpts', 'nit', 'xcutoff', 'nc1c2', 'quadno',
'xsplit']
constants = [getattr(self, name) for name in names]
constants[0] = mod(constants[0], 10)
rindmod.set_constants(*constants) # @UndefinedVariable
def set_constants(self):
if self.xcutoff is None:
trunc_error = 0.1 * self.abseps
self.nc1c2 = max(1, self.nc1c2)
xcut = abs(invnorm(trunc_error / (self.nc1c2 * 2)))
self.xcutoff = max(min(xcut, 8.5), 1.2)
# self.abseps = max(self.abseps- truncError,0);
# self.releps = max(self.releps- truncError,0);
if self.method > 0:
names = ['method', 'xcscale', 'abseps', 'releps', 'coveps',
'maxpts', 'minpts', 'nit', 'xcutoff', 'nc1c2', 'quadno',
'xsplit']
constants = [getattr(self, name) for name in names]
constants[0] = mod(constants[0], 10)
rindmod.set_constants(*constants) # @UndefinedVariable
def initialize(self, speed=None):
"""
Initializes member variables according to speed.
Parameter
---------
speed : scalar integer
defining accuracy of calculations.
Valid numbers: 1,2,...,10
(1=slowest and most accurate,10=fastest, but less accuracy)
Member variables initialized according to speed:
-----------------------------------------------
speed : Integer defining accuracy of calculations.
abseps : Absolute error tolerance.
releps : Relative error tolerance.
covep : Error tolerance in Cholesky factorization.
xcutoff : Truncation limit of the normal CDF
maxpts : Maximum number of function values allowed.
quadno : Quadrature formulae used in integration of Xd(i)
implicitly determining # nodes
"""
if speed is None:
return
self.speed = min(max(speed, 1), 13)
self.maxpts = 10000
self.quadno = r_[1:4] + (10 - min(speed, 9)) + (speed == 1)
if speed in (11, 12, 13):
self.abseps = 1e-1
elif speed == 10:
self.abseps = 1e-2
elif speed in (7, 8, 9):
self.abseps = 1e-2
elif speed in (4, 5, 6):
self.maxpts = 20000
self.abseps = 1e-3
elif speed in (1, 2, 3):
self.maxpts = 30000
self.abseps = 1e-4
if speed < 12:
tmp = max(abs(11 - abs(speed)), 1)
expon = mod(tmp + 1, 3) + 1
self.coveps = self.abseps * ((1.0e-1) ** expon)
elif speed < 13:
self.coveps = 0.1
else:
self.coveps = 0.5
self.releps = min(self.abseps, 1.0e-2)
if self.method == 0:
# This gives approximately the same accuracy as when using
# RINDDND and RINDNIT
# xCutOff= MIN(MAX(xCutOff+0.5d0,4.d0),5.d0)
self.abseps = self.abseps * 1.0e-1
trunc_error = 0.05 * max(0, self.abseps)
self.xcutoff = max(min(abs(invnorm(trunc_error)), 7), 1.2)
self.abseps = max(self.abseps - trunc_error, 0)
def _trdata_cdf(self, data):
'''
Estimate transformation, g, from observed marginal CDF.
Assumption: a Gaussian process, Y, is related to the
non-Gaussian process, X, by Y = g(X).
Parameters
----------
options = options structure defining how the smoothing is done.
(See troptset for default values)
Returns
-------
tr, tr_emp = smoothed and empirical estimate of the transformation g.
The empirical CDF is usually very irregular. More than one local
maximum of the empirical CDF may cause poor fit of the transformation.
In such case one should use a smaller value of GSM or set a larger
variance for GVAR. If X(t) is likely to cross levels higher than 5
standard deviations then the vector param has to be modified. For
example if X(t) is unlikely to cross a level of 7 standard deviations
one can use param = [-7 7 513].
'''
mean = data.mean()
sigma = data.std()
cdf = edf(data.ravel())
Ne = self.ne
nd = len(cdf.data)
if nd > self.ntr and self.ntr > 0:
x0 = np.linspace(cdf.args[Ne], cdf.args[nd - 1 - Ne], self.ntr)
cdf.data = np.interp(x0, cdf.args, cdf.data)
cdf.args = x0
Ne = 0
uu = np.linspace(*self.param)
ncr = len(cdf.data)
ng = len(np.atleast_1d(self.gvar))
if ng == 1:
gvar = self.gvar * np.ones(ncr)
else:
self.gvar = np.atleast_1d(self.gvar)
gvar = np.interp(np.linspace(0, 1, ncr), np.linspace(0, 1, ng),
self.gvar.ravel())
ind = np.flatnonzero(np.diff(cdf.args) > 0) # remove equal points
nd = len(ind)
ind1 = ind[Ne:nd - Ne]
tmp = invnorm(cdf.data[ind])
x = sigma * uu + mean
pp_tr = SmoothSpline(cdf.args[ind1],
tmp[Ne:nd - Ne],
p=self.gsm,
lin_extrap=self.linextrap,
var=gvar[ind1])
tr = TrData(pp_tr(x), x, mean=mean, sigma=sigma)
tr_emp = TrData(tmp, cdf.args[ind], mean=mean, sigma=sigma)
tr_emp.setplotter('step')
if self.chkder:
tr_raw = TrData(tmp[Ne:nd - Ne],
cdf.args[ind1],
mean=mean,
sigma=sigma)
tr = self._check_tr(tr, tr_raw)
if self.plotflag > 0:
tr.plot()
tr_emp.plot()
return tr, tr_emp
def _trdata_lc(self, level_crossings, mean=None, sigma=None):
'''
Estimate transformation, g, from observed crossing intensity.
Assumption: a Gaussian process, Y, is related to the
non-Gaussian process, X, by Y = g(X).
Parameters
----------
mean, sigma : real scalars
mean and standard deviation of the process
**options :
csm, gsm : real scalars
defines the smoothing of the crossing intensity and the
transformation g.
Valid values must be 0<=csm,gsm<=1. (default csm = 0.9 gsm=0.05)
Smaller values gives smoother functions.
param :
vector which defines the region of variation of the data X.
(default [-5, 5, 513]).
monitor : bool
if true monitor development of estimation
linextrap : bool
if true use a smoothing spline with a constraint on the ends to
ensure linear extrapolation outside the range of data. (default)
otherwise use a regular smoothing spline
cvar, gvar : real scalars
Variances for the crossing intensity and the empirical
transformation, g. (default 1)
ne : scalar integer
Number of extremes (maxima & minima) to remove from the estimation
of the transformation. This makes the estimation more robust
against outliers. (default 7)
ntr : scalar integer
Maximum length of empirical crossing intensity. The empirical
crossing intensity is interpolated linearly before smoothing if
the length exceeds ntr. A reasonable NTR (eg. 1000) will
significantly speed up the estimation for long time series without
loosing any accuracy. NTR should be chosen greater than PARAM(3).
(default inf)
Returns
-------
gs, ge : TrData objects
smoothed and empirical estimate of the transformation g.
Notes
-----
The empirical crossing intensity is usually very irregular.
More than one local maximum of the empirical crossing intensity
may cause poor fit of the transformation. In such case one
should use a smaller value of GSM or set a larger variance for GVAR.
If X(t) is likely to cross levels higher than 5 standard deviations
then the vector param has to be modified. For example if X(t) is
unlikely to cross a level of 7 standard deviations one can use
param = [-7 7 513].
Example
-------
>>> import wafo.spectrum.models as sm
>>> import wafo.transform.models as tm
>>> from wafo.objects import mat2timeseries
>>> Hs = 7.0
>>> Sj = sm.Jonswap(Hm0=Hs)
>>> S = Sj.tospecdata() #Make spectrum object from numerical values
>>> S.tr = tm.TrOchi(mean=0, skew=0.16, kurt=0,
... sigma=Hs/4, ysigma=Hs/4)
>>> xs = S.sim(ns=2**16, iseed=10)
>>> ts = mat2timeseries(xs)
>>> tp = ts.turning_points()
>>> mm = tp.cycle_pairs()
>>> lc = mm.level_crossings()
>>> g0, g0emp = lc.trdata(monitor=True) # Monitor the development
>>> g1, g1emp = lc.trdata(gvar=0.5 ) # Equal weight on all points
>>> g2, g2emp = lc.trdata(gvar=[3.5, 0.5, 3.5]) # Less weight on ends
>>> int(S.tr.dist2gauss()*100)
141
>>> int(g0emp.dist2gauss()*100)
380995
>>> int(g0.dist2gauss()*100)
143
>>> int(g1.dist2gauss()*100)
162
>>> int(g2.dist2gauss()*100)
120
g0.plot() # Check the fit.
See also
troptset, dat2tr, trplot, findcross, smooth
NB! the transformated data will be N(0,1)
Reference
---------
Rychlik , I., Johannesson, P., and Leadbetter, M.R. (1997)
"Modelling and statistical analysis of ocean wavedata
using a transformed Gaussian process",
Marine structures, Design, Construction and Safety,
Vol 10, pp 13--47
'''
if mean is None:
mean = level_crossings.mean
if sigma is None:
sigma = level_crossings.sigma
lc1, lc2 = level_crossings.args, level_crossings.data
intensity = level_crossings.intensity
Ne = self.ne
ncr = len(lc2)
if ncr > self.ntr and self.ntr > 0:
x0 = np.linspace(lc1[Ne], lc1[-1 - Ne], self.ntr)
lc1, lc2 = x0, np.interp(x0, lc1, lc2)
Ne = 0
Ner = self.ne
ncr = self.ntr
else:
Ner = 0
ng = len(np.atleast_1d(self.gvar))
if ng == 1:
gvar = self.gvar * np.ones(ncr)
else:
gvar = np.interp(np.linspace(0, 1, ncr), np.linspace(0, 1, ng),
self.gvar)
uu = np.linspace(*self.param)
g1 = sigma * uu + mean
if Ner > 0: # Compute correction factors
cor1 = np.trapz(lc2[0:Ner + 1], lc1[0:Ner + 1])
cor2 = np.trapz(lc2[-Ner - 1::], lc1[-Ner - 1::])
else:
cor1 = 0
cor2 = 0
lc22 = np.hstack((0, cumtrapz(lc2, lc1) + cor1))
if intensity:
lc22 = (lc22 + 0.5 / ncr) / (lc22[-1] + cor2 + 1. / ncr)
else:
lc22 = (lc22 + 0.5) / (lc22[-1] + cor2 + 1)
lc11 = (lc1 - mean) / sigma
lc22 = invnorm(lc22) # - ymean
g2 = TrData(lc22.copy(), lc1.copy(), mean=mean, sigma=sigma)
g2.setplotter('step')
# NB! the smooth function does not always extrapolate well outside the
# edges causing poor estimate of g
# We may alleviate this problem by: forcing the extrapolation
# to be linear outside the edges or choosing a lower value for csm2.
inds = slice(Ne, ncr - Ne) # indices to points we are smoothing over
slc22 = SmoothSpline(lc11[inds], lc22[inds], self.gsm, self.linextrap,
gvar[inds])(uu)
g = TrData(slc22.copy(), g1.copy(), mean=mean, sigma=sigma)
if self.chkder:
tr_raw = TrData(lc22[inds], lc11[inds], mean=mean, sigma=sigma)
g = self._check_tr(g, tr_raw)
if self.plotflag > 0:
g.plot()
g2.plot()
return g, g2
def _trdata_cdf(self, data):
'''
Estimate transformation, g, from observed marginal CDF.
Assumption: a Gaussian process, Y, is related to the
non-Gaussian process, X, by Y = g(X).
Parameters
----------
options = options structure defining how the smoothing is done.
(See troptset for default values)
Returns
-------
tr, tr_emp = smoothed and empirical estimate of the transformation g.
The empirical CDF is usually very irregular. More than one local
maximum of the empirical CDF may cause poor fit of the transformation.
In such case one should use a smaller value of GSM or set a larger
variance for GVAR. If X(t) is likely to cross levels higher than 5
standard deviations then the vector param has to be modified. For
example if X(t) is unlikely to cross a level of 7 standard deviations
one can use param = [-7 7 513].
'''
mean = data.mean()
sigma = data.std()
cdf = edf(data.ravel())
Ne = self.ne
nd = len(cdf.data)
if nd > self.ntr and self.ntr > 0:
x0 = np.linspace(cdf.args[Ne], cdf.args[nd - 1 - Ne], self.ntr)
cdf.data = np.interp(x0, cdf.args, cdf.data)
cdf.args = x0
Ne = 0
uu = np.linspace(*self.param)
ncr = len(cdf.data)
ng = len(np.atleast_1d(self.gvar))
if ng == 1:
gvar = self.gvar * np.ones(ncr)
else:
self.gvar = np.atleast_1d(self.gvar)
gvar = np.interp(np.linspace(0, 1, ncr),
np.linspace(0, 1, ng), self.gvar.ravel())
ind = np.flatnonzero(np.diff(cdf.args) > 0) # remove equal points
nd = len(ind)
ind1 = ind[Ne:nd - Ne]
tmp = invnorm(cdf.data[ind])
x = sigma * uu + mean
pp_tr = SmoothSpline(cdf.args[ind1], tmp[Ne:nd - Ne], p=self.gsm,
lin_extrap=self.linextrap, var=gvar[ind1])
tr = TrData(pp_tr(x), x, mean=mean, sigma=sigma)
tr_emp = TrData(tmp, cdf.args[ind], mean=mean, sigma=sigma)
tr_emp.setplotter('step')
if self.chkder:
tr_raw = TrData(tmp[Ne:nd - Ne], cdf.args[ind1], mean=mean,
sigma=sigma)
tr = self._check_tr(tr, tr_raw)
if self.plotflag > 0:
tr.plot()
tr_emp.plot()
return tr, tr_emp
def _trdata_lc(self, level_crossings, mean=None, sigma=None):
'''
Estimate transformation, g, from observed crossing intensity.
Assumption: a Gaussian process, Y, is related to the
non-Gaussian process, X, by Y = g(X).
Parameters
----------
mean, sigma : real scalars
mean and standard deviation of the process
**options :
csm, gsm : real scalars
defines the smoothing of the crossing intensity and the
transformation g.
Valid values must be 0<=csm,gsm<=1. (default csm = 0.9 gsm=0.05)
Smaller values gives smoother functions.
param :
vector which defines the region of variation of the data X.
(default [-5, 5, 513]).
monitor : bool
if true monitor development of estimation
linextrap : bool
if true use a smoothing spline with a constraint on the ends to
ensure linear extrapolation outside the range of data. (default)
otherwise use a regular smoothing spline
cvar, gvar : real scalars
Variances for the crossing intensity and the empirical
transformation, g. (default 1)
ne : scalar integer
Number of extremes (maxima & minima) to remove from the estimation
of the transformation. This makes the estimation more robust
against outliers. (default 7)
ntr : scalar integer
Maximum length of empirical crossing intensity. The empirical
crossing intensity is interpolated linearly before smoothing if
the length exceeds ntr. A reasonable NTR (eg. 1000) will
significantly speed up the estimation for long time series without
loosing any accuracy. NTR should be chosen greater than PARAM(3).
(default inf)
Returns
-------
gs, ge : TrData objects
smoothed and empirical estimate of the transformation g.
Notes
-----
The empirical crossing intensity is usually very irregular.
More than one local maximum of the empirical crossing intensity
may cause poor fit of the transformation. In such case one
should use a smaller value of GSM or set a larger variance for GVAR.
If X(t) is likely to cross levels higher than 5 standard deviations
then the vector param has to be modified. For example if X(t) is
unlikely to cross a level of 7 standard deviations one can use
param = [-7 7 513].
Example
-------
>>> import wafo.spectrum.models as sm
>>> import wafo.transform.models as tm
>>> from wafo.objects import mat2timeseries
>>> Hs = 7.0
>>> Sj = sm.Jonswap(Hm0=Hs)
>>> S = Sj.tospecdata() #Make spectrum object from numerical values
>>> S.tr = tm.TrOchi(mean=0, skew=0.16, kurt=0,
... sigma=Hs/4, ysigma=Hs/4)
>>> xs = S.sim(ns=2**16, iseed=10)
>>> ts = mat2timeseries(xs)
>>> tp = ts.turning_points()
>>> mm = tp.cycle_pairs()
>>> lc = mm.level_crossings()
>>> g0, g0emp = lc.trdata(monitor=True) # Monitor the development
>>> g1, g1emp = lc.trdata(gvar=0.5 ) # Equal weight on all points
>>> g2, g2emp = lc.trdata(gvar=[3.5, 0.5, 3.5]) # Less weight on ends
>>> int(S.tr.dist2gauss()*100)
141
>>> int(g0emp.dist2gauss()*100)
380995
>>> int(g0.dist2gauss()*100)
143
>>> int(g1.dist2gauss()*100)
162
>>> int(g2.dist2gauss()*100)
120
g0.plot() # Check the fit.
See also
troptset, dat2tr, trplot, findcross, smooth
NB! the transformated data will be N(0,1)
Reference
---------
Rychlik , I., Johannesson, P., and Leadbetter, M.R. (1997)
"Modelling and statistical analysis of ocean wavedata
using a transformed Gaussian process",
Marine structures, Design, Construction and Safety,
Vol 10, pp 13--47
'''
if mean is None:
mean = level_crossings.mean
if sigma is None:
sigma = level_crossings.sigma
lc1, lc2 = level_crossings.args, level_crossings.data
intensity = level_crossings.intensity
Ne = self.ne
ncr = len(lc2)
if ncr > self.ntr and self.ntr > 0:
x0 = np.linspace(lc1[Ne], lc1[-1 - Ne], self.ntr)
lc1, lc2 = x0, np.interp(x0, lc1, lc2)
Ne = 0
Ner = self.ne
ncr = self.ntr
else:
Ner = 0
ng = len(np.atleast_1d(self.gvar))
if ng == 1:
gvar = self.gvar * np.ones(ncr)
else:
gvar = np.interp(np.linspace(0, 1, ncr),
np.linspace(0, 1, ng), self.gvar)
uu = np.linspace(*self.param)
g1 = sigma * uu + mean
if Ner > 0: # Compute correction factors
cor1 = np.trapz(lc2[0:Ner + 1], lc1[0:Ner + 1])
cor2 = np.trapz(lc2[-Ner - 1::], lc1[-Ner - 1::])
else:
cor1 = 0
cor2 = 0
lc22 = np.hstack((0, cumtrapz(lc2, lc1) + cor1))
if intensity:
lc22 = (lc22 + 0.5 / ncr) / (lc22[-1] + cor2 + 1. / ncr)
else:
lc22 = (lc22 + 0.5) / (lc22[-1] + cor2 + 1)
lc11 = (lc1 - mean) / sigma
lc22 = invnorm(lc22) # - ymean
g2 = TrData(lc22.copy(), lc1.copy(), mean=mean, sigma=sigma)
g2.setplotter('step')
# NB! the smooth function does not always extrapolate well outside the
# edges causing poor estimate of g
# We may alleviate this problem by: forcing the extrapolation
# to be linear outside the edges or choosing a lower value for csm2.
inds = slice(Ne, ncr - Ne) # indices to points we are smoothing over
slc22 = SmoothSpline(lc11[inds], lc22[inds], self.gsm, self.linextrap,
gvar[inds])(uu)
g = TrData(slc22.copy(), g1.copy(), mean=mean, sigma=sigma)
if self.chkder:
tr_raw = TrData(lc22[inds], lc11[inds], mean=mean, sigma=sigma)
g = self._check_tr(g, tr_raw)
if self.plotflag > 0:
g.plot()
g2.plot()
return g, g2
def initialize(self, speed=None):
'''
Initializes member variables according to speed.
Parameter
---------
speed : scalar integer
defining accuracy of calculations.
Valid numbers: 1,2,...,10
(1=slowest and most accurate,10=fastest, but less accuracy)
Member variables initialized according to speed:
-----------------------------------------------
speed : Integer defining accuracy of calculations.
abseps : Absolute error tolerance.
releps : Relative error tolerance.
covep : Error tolerance in Cholesky factorization.
xcutoff : Truncation limit of the normal CDF
maxpts : Maximum number of function values allowed.
quadno : Quadrature formulae used in integration of Xd(i)
implicitly determining # nodes
'''
if speed is None:
return
self.speed = min(max(speed, 1), 13)
self.maxpts = 10000
self.quadno = r_[1:4] + (10 - min(speed, 9)) + (speed == 1)
if speed in (11, 12, 13):
self.abseps = 1e-1
elif speed == 10:
self.abseps = 1e-2
elif speed in (7, 8, 9):
self.abseps = 1e-2
elif speed in (4, 5, 6):
self.maxpts = 20000
self.abseps = 1e-3
elif speed in (1, 2, 3):
self.maxpts = 30000
self.abseps = 1e-4
if speed < 12:
tmp = max(abs(11 - abs(speed)), 1)
expon = mod(tmp + 1, 3) + 1
self.coveps = self.abseps * ((1.0e-1) ** expon)
elif speed < 13:
self.coveps = 0.1
else:
self.coveps = 0.5
self.releps = min(self.abseps, 1.0e-2)
if self.method == 0:
# This gives approximately the same accuracy as when using
# RINDDND and RINDNIT
# xCutOff= MIN(MAX(xCutOff+0.5d0,4.d0),5.d0)
self.abseps = self.abseps * 1.0e-1
trunc_error = 0.05 * max(0, self.abseps)
self.xcutoff = max(min(abs(invnorm(trunc_error)), 7), 1.2)
self.abseps = max(self.abseps - trunc_error, 0)